Example Application of the b-Core: Buyer Coalition Problem with Beliefs - Belief-based Stability in Coalition Formation with Uncertainty - page 130

Proof If all agents' beliefs are accurate, then for any two coalition
structure CS and CS and any two agent i and j ,wehave

bel i ( b j|CS j b j|CS ) ↔ b j|CS j b j|CS .

This implies that the fulfilment of condition 1) in Definition 5.5 also
implies the fulfilment of condition 2). By Definitions 5.4 and 5.5, the
core of a NTU-Buyer game is the same as the b-core in this case.

In general, we have the following result.

Theorem 5.3 Given two NTU-Buyer games

g = i ) ,B g = N,G, ( i ) ,B , N,G, (

we have

b-core( g ) b-core( g ) ⊆

if B is more accurate than B .

Proof Consider an objection in the game g against a coalition struc-
ture S 1 . By Definition 5.5, there exists an alternative coalition struc-
ture S 2 , a coalition C ,abid b , a coalition C ∈ S 2 and an agent i ∈ C

such that, for each agent k ∈ C ,wehave

bel i ( b 1 k b ( S 1 )) .

There are only two cases to consider here. First, if both the belief bel i
and bel i are accurate, then the result of the two games will be the same,
meaning that either they are both valid objections, or both invalid.
Second, if bel i is accurate but bel i is not, then the latter objection
would be invalid. Thus, we see that any valid objection for the game
g is also a valid objection for the game g ,hence

b-core( g ) . b-core( g ) ⊆